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Suppose that C is a fusion C*-cateogry and that A is an irreducible Frobenius algebra object in C, is there always a Frobenius algebra A' isomorphic to A such that A' is a Longo Q-system (that is the coproduct on A' is an isometry)? In other words, wlog can one assume that the coproduct is the * of the product?

The motivation of this question is to understand whether the theory of irreducible Frobenius algebra objects in monoidal *-categories actually agrees on the nose with the theory of irreducible subfactors or whether there's a small loophole.

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  • $\begingroup$ So here's a sketch of how I was thinking of proving this before I thought I had a counterexample. Look at the 2-category of 1-1, 1-A, A-1, and A-A bimodules. We want to prove that this is a 2-C*-category and then by work of Longo we could conclude that A is a Q-system. But all those bimodule categories have a forgetful functor to the original category, so you just use the original *-structure to inherit a *-structure on the 2-category. You then check that this is a C* structure and you're done. $\endgroup$
    – Noah Snyder
    Commented Nov 4, 2010 at 22:30

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This is a comment on Noah's answer, posted as an answer due to lack of reputation. The semion MTC is inequivalent to Vec(Z/2) as a fusion category; it is the other rank two fusion category. Confusingly, there is a change in sign in one of the F-matrices AND a change in sign in the pivotal structure which gives unitarity; the two occur simultaneously in most diagrams.

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  • $\begingroup$ Ah very good. Right, I think at one point a long time ago I worked this out and then I forgot. There are 4 *-fusion categories here, you can move between them by changing * structure or by changing associator. If you only do one or the other then you negate dimensions, but if you do both you keep dimensions the same. So only 2 of them are positive definite: the usual associator and usual *-structure or unusual associator and unusual *-structure. In particular, 1+X is only an algebra object in the former case (because the module category is Vec giving a fiber functor). $\endgroup$
    – Noah Snyder
    Commented Nov 4, 2010 at 21:45
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This answer is wrong, the semion category has a nontrivial associator and so 1+X is not an algebra there. See Tobias's answer.

I think the answer to this question is "no." Below I explain a counterexample.

Consider the fusion category Vec(Z/2) with two objects 1 and X. The object 1+X has a natural Frobenius algebra structure (just think of it as the group ring C[Z/2]). However, Vec(Z/2) has two different *-structures: the usual *-structure where X is real (aka orthogonal) and the *-structure where X is pseudoreal (aka quaternionic aka symplectic). In the latter case 1+X can't have a Q-system structure by remark 3 on page 30 of Mueger's From Subfactors to Categories and Topology I which explains that Q-systems are always real.

The tricky point in the above is checking that Vec(Z/2) really does have a second *-structure. I worked this out diagrammatically, but it can also be realized by looking atU_q(sl_2) when q is a primitive 6th root of unity since spin 1/2 reps are pseudoreal. This tensor *-category is called the called the "semion" theory in Section 5.3.1. of Rowell-Strong-Wang's On classification of modular tensor categories where they note that the nontrivial object has "Frobenius-Schur indicator -1", in other words it's pseudoreal.

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  • $\begingroup$ With this second -structure, does TT >= 0 hold for any morphism? If not, your counter-example would be ruled out for C*-categories. $\endgroup$
    – pasquale zito
    Commented Oct 7, 2010 at 22:34
  • $\begingroup$ Yes, I think TT>=0, unless I somehow made a mistake. In particular in the Rowell-Strong-Wang paper they're only considering unitary modular tensor categories, and unitary means C. $\endgroup$
    – Noah Snyder
    Commented Oct 7, 2010 at 22:49
  • $\begingroup$ Looks like Tobias is right above. $\endgroup$
    – Noah Snyder
    Commented Nov 4, 2010 at 21:42
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This is not a full answer. The answer is yes for weakly group-theoretical fusion categories. The question is equivalent to: let C be a unitary fusion category, does every indecomposable C-module category admit a compatible unitary structure (see GMR, for all definitions). In Theorem 5.20, we prove that a weakly group-theoretical fusion category admits a unique unitary structure and every indecomposable module category also admits a unique compatible unitary structure.

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